Combine shallot, half-and-half, parsley, egg, and 1/3 cup Italian breadcrumbs in a mixing bowl and stir together.

Add ham and chicken; season with salt, pepper, garlic powder, and paprika. Mix all ingredients together just until uniform.

Add remaining Italian breadcrumbs and panko to a shallow dish and stir to combine. Roll meat mixture into 12 balls and stuff a Swiss cheese cube into the center. Roll each ball in breadcrumbs until evenly coated; place on a plate or baking sheet. Freeze meatballs for 10 minutes.

Meanwhile, preheat the oven to 400°F (200°C) and set a wire rack inside a rimmed baking sheet.

Heat oil in a large skillet over medium-high heat. Add meatballs and cook until browned on all sides, 5 to 10 minutes. Transfer meatballs to the prepared wire rack.

Bake meatballs in the preheated oven until an instant read thermometer, inserted near the center, reads 165°F (74°C), about 10 minutes.

Meanwhile, make Dijon sauce. Wipe skillet clean and melt butter over medium heat. Whisk in flour and cook 1 minute. Whisk in chicken broth and half-and-half. Stir in mustard, salt, and pepper and bring to a simmer. Reduce heat and simmer until sauce is rich and creamy, about 5 minutes. Remove from heat and stir in lemon juice.